Find the value of 1 - - - -2 - -3 - -n-1- If -k - cos2-k-n - i sin2-k-n-K - 0-1-2-n-1

1 + α + α^2 + α^3 +................... α^n 1 Sum of the GP = 1(1 a^n)/1 a When k =n Given α^n = cos2πk/n + i sin2nk/n = 1 ...

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Find the value of 1 + $α$ + $α^{2}$ + $α^{3}$ +.............. $α^{n - 1}$ . If $α^{k}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 π k}{n}$ ,

K = 0,1,2,..........n-1

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Solution

1 + $α$ + $α^{2}$ + $α^{3}$ +................... $α^{n - 1}$

Sum of the GP = $\frac{1 (1 - a^{n})}{1 - a}$

When k =n

Given $α^{n}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 n k}{n}$ = 1

Sum of the GP = $\frac{1 (1 - 1)}{1 - a}$ = 0

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1 + $α$ + $α^{2}$ + $α^{3}$ +................... $α^{n - 1}$

Sum of the GP = $\frac{1 (1 - a^{n})}{1 - a}$

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